Methods for Solving L R -Type Pythagorean Fuzzy Linear Programming Problems with Mixed Constraints

A Pythagorean fuzzy set is the superset of fuzzy and intuitionistic fuzzy sets, respectively. Yager proposed the concept of Pythagorean fuzzy sets in which he relaxed the condition that sum of square of both membership degree and nonmembership degree of an element of a set must not be greater than 1. This paper introduces two new techniques to solve L R -type fully Pythagorean fuzzy linear programming problems with mixed constraints having unrestricted L R -type Pythagorean fuzzy numbers as variables and parameters by introducing unknown variables and using a ranking function. Furthermore, we show the equivalence of both the proposed methods and compare the solutions obtained by the two techniques. Besides this, we solve an already existing practical model using proposed techniques and compare the result.

On Robust Saddle-Point Criterion in Optimization Problems with Curvilinear Integral Functionals

In this paper, we introduce a new class of multi-dimensional robust optimization problems (named (P)) with mixed constraints implying second-order partial differential equations (PDEs) and inequations (PDIs). Moreover, we define an auxiliary (modified) class of robust control problems (named (P)(b¯,c¯)), which is much easier to study, and provide some characterization results of (P) and (P)(b¯,c¯) by using the notions of normal weak robust optimal solution and robust saddle-point associated with a Lagrange functional corresponding to (P)(b¯,c¯). For this aim, we consider path-independent curvilinear integral cost functionals and the notion of convexity associated with a curvilinear integral functional generated by a controlled closed (complete integrable) Lagrange 1-form.

On a Class of Second-Order PDE&PDI Constrained Robust Modified Optimization Problems

In this paper, by using scalar multiple integral cost functionals and the notion of convexity associated with a multiple integral functional driven by an uncertain multi-time controlled second-order Lagrangian, we develop a new mathematical framework on multi-dimensional scalar variational control problems with mixed constraints implying second-order partial differential equations (PDEs) and inequations (PDIs). Concretely, we introduce and investigate an auxiliary (modified) variational control problem, which is much easier to study, and provide some equivalence results by using the notion of a normal weak robust optimal solution.

Investigation of Optimization Algorithms for Neural Network Solutions of Optimal Control Problems with Mixed Constraints

In this paper, we consider the problem of selecting the most efficient optimization algorithm for neural network approximation—solving optimal control problems with mixed constraints. The original optimal control problem is reduced to a finite-dimensional optimization problem by applying the necessary optimality conditions, the Lagrange multiplier method and the least squares method. Neural network approximation models are presented for the desired control functions, trajectory and conjugate factors. The selection of the optimal weight coefficients of the neural network approximation was carried out using the gravitational search algorithm and the basic particle swarm algorithm and the genetic algorithm. Computational experiments showed that evolutionary optimization algorithms required the smallest number of iterations for a given accuracy in comparison with the classical gradient optimization method; however, the multi-agent optimization methods were performed later for each operation. As a result, the genetic algorithm showed a faster convergence rate relative to the total execution time.

Necessary optimality conditions in nonsmooth semi-infinite multiobjective optimization under metric subregularity

We consider nonsmooth semi-infinite multiobjective optimization problems under mixed constraints, including infinitely many mixed constraints by using Clarke subdifferential. Semi-infinite programming (SIP) is the minimization of many scalar objective functions subject to a possibly infinite system of inequality or/and equality constraints. SIPs have been proved to be very important in optimization and applications. Semi-infinite programming problems arise in various fields of engineering such as control systems design, decision making under competition, and multiobjective optimization. There is extensive literature on standard semi-infinite programming problems. The investigation of optimality conditions for these problems is always one of the most attractive topics and has been studied extensively in the literature. In our work, we study optimality conditions for weak efficiency of a multiobjective semi-infinite optimization problem under mixed constraints including infinitely many of both equality and inequality constraints in terms of Clarke subdifferential. Our conditions are the form of the Karush-Kuhn-Tucker (KKT) multiplier. To the best of our knowledge, only a few papers are dealing with optimality conditions for SIPs subject to mixed constraints. By the Pshenichnyi-Levin-Valadire (PLV) property and the directional metric subregularity, we introduce a type of Mangasarian-Fromovitz constraint qualification (MFCQ). Then we show that (MFCQ) is a sufficient condition to guarantee the extended Abadie constraint qualification (ACQ) to satisfy. In our constraint qualifications, all functions are nonsmooth and the number of constraints is not necessarily finite. In our paper, we do not need the involved functions: convexity and differentiability. Later, we apply the extended Abadie constraint qualification to get the KKT multipliers for weak efficient solutions of SIP. Many examples are provided to illustrate some advantages of our results. The paper is organized as follows. In Section Preliminaries, we present our basic definitions of nonsmooth and convex analysis. Section Main Results prove necessary conditions for the weakly efficient solution in terms of the Karush-Kuhn-Tucker mult iplier rule with the help of some constraint qualifications.